Problem 70 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. $$ m=-9, \quad(-4,-7) $$

Step-by-Step Solution

Answer

Answer: The equation of the line in the slope-intercept form is given by: \(y = -9x - 43\).

Step 1: Write down the given information

We are given: Slope \(m = -9\) Point \((-4, -7)\)

Step 2: Insert the given information into the slope-intercept form equation

Substitute the slope \(m\) and the coordinates of the given point \((-4,-7)\) into the equation \(y=mx+b\): $$ -7 = -9(-4) + b $$

Step 3: Solve for the y-intercept, b

Now we can solve for \(b\): $$ -7 = 36 + b $$ Subtract 36 from both sides to find \(b\): $$ b = -7 - 36 $$ So, $$ b = -43 $$

Step 4: Insert the slope and y-intercept into the equation

Now that we have found \(b\), we can write the complete equation of the line in slope-intercept form by inserting the values of \(m\) and \(b\): $$ y = -9x - 43 $$ The equation of the line in the slope-intercept form is given by: $$ y = -9x - 43 $$

Key Concepts

Linear Equations

Linear equations describe relationships between two variables, usually in the form of a line on a graph.
They can be expressed in several forms, but one popular way is through the slope-intercept form. This form is useful because it clearly demonstrates important features of the line, such as its slope and y-intercept.

• Slope-Intercept Form: The general formula is \( y = mx + b \).
• Here, \( m \) represents the slope, while \( b \) denotes the y-intercept.

Linear equations help us predict one variable given the known value of the other. Because they're straight lines, they make predictions straightforward and reliable.
All you need are two pieces of information: the slope and the y-intercept.

Slope of a Line

The slope of a line, represented by the letter \( m \), indicates the steepness and direction of the line.
It is calculated as the "rise over run," which means the change in the y-value over the change in the x-value between two points.

• Positive Slope: As you move from left to right, the line ascends.
• Negative Slope: As you move from left to right, the line descends.
• Zero Slope: The line is perfectly horizontal.
• Undefined Slope: The line is perfectly vertical.

In the given exercise, the slope \( m \) is \( -9 \), which means the line is steeply descending.
This negative slope guides how steeply the line will decline as x increases.

y-intercept

The y-intercept is a crucial point where the line crosses the y-axis on a graph. This gives us a specific point where our line intersects the vertical axis and is denoted by \( b \) in the slope-intercept form \( y = mx + b \).
It tells us the value of y when x equals 0 and is an essential starting point for plotting the line.
To find the y-intercept in an equation, you can substitute the slope and a point on the line into the slope-intercept form and then solve for \( b \).
In the example provided, after substituting the slope and the coordinates from the point \((-4,-7)\), it was determined that \( b = -43 \).

• Practical Use: Knowing the y-intercept allows you to easily graph the line by starting from the y-axis and using the slope to plot the line correctly.

Understanding the y-intercept helps provide context to the line, making it easier to visualize its behavior on a graph.